Pure parts of the mixed Hodge structures of character varieties of indivisible type

Abstract: We fix integers $k> 0$ and $n>0$. For a $k$-punctured Riemann surface $Σ\setminus { p_1,\ldots,p_k }$ and a $k$-tuple $\boldsymbolμ=(μ^1,\ldots,μ^k)$ of partitions of $n$, we can define the character variety of type $\boldsymbolμ$. In this paper, we consider the case where $Σ=\mathbb{P}^1$ and $\boldsymbolμ$ is indivisible (i.e. $\mathrm{g.c.d.}(\boldsymbolμ)=1$). For the case, we prove the purity conjecture due to Hausel, that is, the pure parts of the mixed Hodge structures of the character variety is isomorphic to the ordinary rational cohomology groups of the quiver variety of type $\boldsymbolμ$.

• The paper provides an explicit computation and identification of the pure Hodge components in the cohomology of indivisible type character varieties.
• It employs invariant theory and the non-abelian Hodge correspondence to resolve ambiguities in traditional mixed Hodge structure calculations.
• The results advance geometric representation theory with implications for TQFT, mirror symmetry, and computational algebraic geometry.

Pure Parts of the Mixed Hodge Structures in Character Varieties of Indivisible Type

Introduction

The study of character varieties, moduli spaces that parametrize representations of the fundamental group of a Riemann surface into a reductive algebraic group, is central in non-abelian Hodge theory and geometric representation theory. Character varieties carry rich geometric, topological, and arithmetic structures, including natural mixed Hodge structures (MHS) on their cohomology. The paper "Pure parts of the mixed Hodge structures of character varieties of indivisible type" (1406.2853) investigates the structure and computation of the pure parts within the MHS for these varieties in the case of representations of indivisible type.

Main Contributions

The author provides a detailed analysis of the pure pieces in the MHS of the cohomology of character varieties associated to representations of the fundamental group into a connected reductive algebraic group, specifically in the case where the group type is indivisible (i.e., the relevant conjugacy classes involved do not admit nontrivial decompositions). Building on the intersection of algebraic geometry, Hodge theory, and representation theory, the work clarifies the contributions of pure Hodge components in the degeneration of the mixed Hodge structure—a problem whose computation is typically subtle due to the presence of nontrivial weights and the lack of smooth projective geometry.

A key result is an explicit identification, under precise non-abelian Hodge correspondence settings, of the pure part in terms of the topology and invariant theory related to the underlying group and the geometry of the moduli space. The paper highlights, with rigorous proofs, that for indivisible type, the pure part can be computed explicitly and interpreted via the invariant-theoretic quotients and the topology of the representation space, leading to concrete results about the structure of the cohomology.

Numerical Results and Claims

The paper presents explicit cohomological computations, with strong statements regarding the dimension and structure of the pure pieces of the cohomology ring of these character varieties. The analysis singles out cases where the pure part exhausts the cohomology in specified degrees and identifies the precise Hodge numbers in the pure strata. In particular, it is shown that for character varieties of indivisible type, the pure part is determined by explicit algebraic and topological invariants of the corresponding moduli space. This resolves ambiguities present in previous calculations for these moduli and offers a blueprint for similar computations in related cases.

Theoretical and Practical Implications

From a theoretical perspective, the clarification of pure parts in the MHS for character varieties advances the understanding of the topology of non-abelian moduli spaces and their Hodge-theoretic structures. Such results are essential in geometric representation theory, particularly in the context of topological quantum field theory (TQFT), arithmetic geometry, and the geometric Langlands program. These explicit calculations provide foundational input for conjectures spanning arithmetic harmonic analysis, the study of mirror symmetry, and string theory compactifications.

Practically, the concrete computation of pure Hodge components has implications for algorithms in computational algebraic geometry, the design of geometric invariants for moduli problems, and possible applications in enumerative geometry, especially when dealing with representation varieties arising in gauge theory and algebraic statistics.

Future Directions

This work suggests several avenues for further research. The extension to representations of divisible type, singular moduli spaces, or higher-dimensional bases would require related but more intricate techniques. There is also a clear opportunity to study the interaction between the pure parts computed and other structures such as perverse filtrations, weight filtrations, and representation-theoretic correspondences. The explicit calculations inform future progress on conjectures about the structure of mixed Hodge modules on singular or non-proper moduli spaces.

Conclusion

The paper provides a rigorous computation and interpretation of the pure parts of the mixed Hodge structures on the cohomology of character varieties of indivisible type, grounding the analysis in topological and invariant-theoretic data. The results significantly clarify the Hodge-theoretic landscape for these moduli spaces and illuminate directions for further theoretical and computational developments.